1. **Problem Statement:** We have a dilation centered at the origin that transforms Figure A into Figure B. We need to find the scale factor of this dilation. 2. **Understanding Dilation:** A dilation centered at the origin scales every point $(x,y)$ of the original figure by a scale factor $k$ to a new point $(kx, ky)$. 3. **Finding the Scale Factor:** To find $k$, pick a corresponding point from Figure A and Figure B. 4. **Example Points:** Suppose a point on Figure A is $(x_1, y_1)$ and its image on Figure B is $(x_2, y_2)$. 5. **Calculate $k$:** Since the dilation is centered at the origin, the scale factor is $$ k = \frac{x_2}{x_1} = \frac{y_2}{y_1} $$ 6. **Using Coordinates:** If Figure A has a vertex at $(4, 2)$ and Figure B has the corresponding vertex at $(2, 1)$, then $$ k = \frac{2}{4} = \frac{1}{2} $$ 7. **Simplify:** The scale factor is $\frac{1}{2}$. **Final answer:** The scale factor of the dilation is $\frac{1}{2}$.